在理想条件下，找到二分类器最优学习速率的理论下界。
Minimax learning rates for estimating binary classifiers under margin conditions
- 基于几何间隔条件推导泛化误差下界
- 对三类函数边界实现接近O(n⁻¹)的快速学习率
- 适用于理论分析与高精度分类场景
我们研究了决策边界由水平函数描述的二分类问题，数据分布满足几何间隔条件。本文的核心创新在于，在几乎普遍存在的几何间隔条件下，推导出广泛函数类上最坏情况学习速率的下界——这一设置虽在实践中普遍存在，但理论分析极具挑战性，尤其在无噪声情形下，下界更难建立。我们的通用结果涵盖多种决策边界函数类：对于Barron-正则、Hölder连续及强间隔下的凸Lipschitz函数，均识别出接近O(n⁻¹)的最优学习速率，其中n为样本数量。
原文摘要 · Abstract (English)
We study classification problems using binary estimators where the decision boundary is described by horizon functions and where the data distribution satisfies a geometric margin condition. A key novelty of our work is the derivation of lower bounds for the worst-case learning rates over broad classes of functions, under a geometric margin condition -- a setting that is almost universally satisfied in practice, but remains theoretically challenging. Moreover, we work in the noiseless setting, where lower bounds are particularly hard to establish. Our general results cover, in particular, classification problems with decision boundaries belonging to several classes of functions: for Barron-regular functions, Hölder-continuous functions, and convex-Lipschitz functions with strong margins, we identify optimal rates close to the fast learning rates of $\mathcal{O}(n^{-1})$ for $n \in \mathbb{N}$ samples.
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